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Mirrors > Home > MPE Home > Th. List > arwhoma | Structured version Visualization version GIF version |
Description: An arrow is contained in the hom-set corresponding to its domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwhoma.h | ⊢ 𝐻 = (Homa‘𝐶) |
Ref | Expression |
---|---|
arwhoma | ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwrcl.a | . . . . . . 7 ⊢ 𝐴 = (Arrow‘𝐶) | |
2 | arwhoma.h | . . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶) | |
3 | 1, 2 | arwval 17988 | . . . . . 6 ⊢ 𝐴 = ∪ ran 𝐻 |
4 | 3 | eleq2i 2826 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ ∪ ran 𝐻) |
5 | 4 | biimpi 215 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ∪ ran 𝐻) |
6 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
7 | 1 | arwrcl 17989 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝐶 ∈ Cat) |
8 | 2, 6, 7 | homaf 17975 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → 𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V)) |
9 | ffn 6713 | . . . . 5 ⊢ (𝐻:((Base‘𝐶) × (Base‘𝐶))⟶𝒫 (((Base‘𝐶) × (Base‘𝐶)) × V) → 𝐻 Fn ((Base‘𝐶) × (Base‘𝐶))) | |
10 | fnunirn 7247 | . . . . 5 ⊢ (𝐻 Fn ((Base‘𝐶) × (Base‘𝐶)) → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))) | |
11 | 8, 9, 10 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ ∪ ran 𝐻 ↔ ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧))) |
12 | 5, 11 | mpbid 231 | . . 3 ⊢ (𝐹 ∈ 𝐴 → ∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧)) |
13 | fveq2 6887 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝐻‘〈𝑥, 𝑦〉)) | |
14 | df-ov 7406 | . . . . . 6 ⊢ (𝑥𝐻𝑦) = (𝐻‘〈𝑥, 𝑦〉) | |
15 | 13, 14 | eqtr4di 2791 | . . . . 5 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐻‘𝑧) = (𝑥𝐻𝑦)) |
16 | 15 | eleq2d 2820 | . . . 4 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝐹 ∈ (𝐻‘𝑧) ↔ 𝐹 ∈ (𝑥𝐻𝑦))) |
17 | 16 | rexxp 5839 | . . 3 ⊢ (∃𝑧 ∈ ((Base‘𝐶) × (Base‘𝐶))𝐹 ∈ (𝐻‘𝑧) ↔ ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
18 | 12, 17 | sylib 217 | . 2 ⊢ (𝐹 ∈ 𝐴 → ∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦)) |
19 | id 22 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ (𝑥𝐻𝑦)) | |
20 | 2 | homadm 17985 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (doma‘𝐹) = 𝑥) |
21 | 2 | homacd 17986 | . . . . . 6 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → (coda‘𝐹) = 𝑦) |
22 | 20, 21 | oveq12d 7421 | . . . . 5 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → ((doma‘𝐹)𝐻(coda‘𝐹)) = (𝑥𝐻𝑦)) |
23 | 19, 22 | eleqtrrd 2837 | . . . 4 ⊢ (𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
24 | 23 | rexlimivw 3152 | . . 3 ⊢ (∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
25 | 24 | rexlimivw 3152 | . 2 ⊢ (∃𝑥 ∈ (Base‘𝐶)∃𝑦 ∈ (Base‘𝐶)𝐹 ∈ (𝑥𝐻𝑦) → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
26 | 18, 25 | syl 17 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐹 ∈ ((doma‘𝐹)𝐻(coda‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 𝒫 cpw 4600 〈cop 4632 ∪ cuni 4906 × cxp 5672 ran crn 5675 Fn wfn 6534 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 domacdoma 17965 codaccoda 17966 Arrowcarw 17967 Homachoma 17968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-1st 7969 df-2nd 7970 df-doma 17969 df-coda 17970 df-homa 17971 df-arw 17972 |
This theorem is referenced by: arwdm 17992 arwcd 17993 arwhom 17996 arwdmcd 17997 coapm 18016 |
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